Lattice potential

Surface Stresses and Edge Energies. Some surface tension values, that is, values of the surface stress t, are included in Table VII-2. These are obtained by applying Eq. Vll-5 to the appropriate lattice sums. The calculation is very sensitive to the form of the lattice potential. Earlier calculations have given widely different results, including negative r [43, 51, 52]. 

The smoothing terms have a thermodynamic basis, because they are related to surface gradients in chemical potential, and they are based on linear rate equations. The magnitude of the smoothing terms vary with different powers of a characteristic length, so that at large scales, the EW term should predominate, while at small scales, diffusion becomes important. The literature also contains non-linear models, with terms that may represent the lattice potential or account for step growth or diffusion bias, for example. 

The lattice potential V (or the cluster potential) is written as a sum of atomic potentials V, centered on the lattice sites i. 

In a subsequent study, NoorBatcha et al. varied the valence-force parameters used for the lattice interactions to evaluate the effect of the vibrational properties of the crystal on diffusion characteristics. Using three sets of lattice potential parameters, they determined a range of effective activation barriers for diffusion of 3.63 kcal/mole to 7.47 kcal/mole on the Si(001) surface. This range encompasses the experimental estimate of 4.6 kcal/mole for the Si(lll) surface, and further suggests this value as the more accurate experimental estimate. 

Fig. 2-4. Lattice potential energy and electron energy bands in crystals IB s inner band FB s frontier band.

G. Radi, Complex Lattice Potentials in Electron Diffraction Calculated for a number of Crystals, Acta Cryst., A26,41-56,1970. 

Order-disorder-type ferroeiectrics where a discrete symmetry group is broken due to the ordering of the ions in a rigid lattice potential (e.g., KH2PO4). 

CS planes using conventional TEM methods (e.g. Merritt and Hyde 1973) and by calculations using lattice potential models (Catlow et al 1978, Cormack et al... 

An interesting application of the fast-forward protocol is site-to-site population transfer of particles in a BEC confined on an optical lattice. As shown below, the features of that protocol differ somewhat between continuous and lattice systems. We consider a BEC subjected to an external potential that is a superposition of a lattice potential and a harmonic potential, with the representation... 

In Eq. (3.106), A/2 is the period of the lattice potential, Xq is the x-coordinate of the bottom of the harmonic potential and f/ determines the tightness of the transverse confinement of a particle in the lattice. Using the tight-binding approximation, the condensate order parameter is [80, 81]... 

Next, we examine the term i2. In a gas-like single segment approximation, this term can be replaced by 1212. The molecular conformation statistics are independent of each other. This might be due to the fact that in the absence of a three-dimensional lattice-potential, nematic shifts of neighboring segments are very likely to occur. In this approximation the configuration does not depend on which individual pair of molecules k, 1 is picked out The molecular structure factor is independent of the indexes k and L Hence 1 inter, d can be written as... 

Figure 5-5. a) Point defect potential in an ionic crystal superposition of the periodic lattice potential and the individual defect potential valley, b) Change of potential with time after a defect... 

The influence of a commensurate lattice potential on a free density wave is considered in section 5. The full finite temperature renormalization group flow equation for this sine-Gordon type model are derived and resulting phase diagram is discussed. Furthermore a qualitative picture of the combined effect of disorder and a commensurate lattice potential at zero temperature is presented in section 6, including the phase diagram. 

The last term in (4), Tiw, includes the influence of a harmonic lattice potential. This term will be discussed section 5 in greater detail. 

If the wave length A of the CDW modulation is commensurate with the period a (= 7r, due to dimensionless units) of the underlying lattice such that nX = qa with integers n and q, the umklapp term —2n(w/K)cos(qf) appears in the Hamiltonian [23]. Therefore we switch on the lattice potential o/0 now. In this section we consider the case u = 0, which leads to the sine-Gordon type model ... 

Fig. 8. Qualitative zero temperature phase diagram for a system with commensurate lattice potential and small disorder. One has to distinguish two cases (i) < A ...

The combined effect of disorder and the lattice potential on the zero temperature phase diagram, i.e., the competition between unpinning (Anderson) and lock-in (Mott) transition, is still controversially discussed [41, 14] and cannot be explained by the RG-results presented here, since both perturbations become relevant for small K. However, using Imry-Ma arguments one finds, that as soon as If is below one of the two critical values (for the unpinning and lock-in transition) the disorder dominates the lattice potential and only two phases exist. This is in contrast to the proposed existence of a so-called intermediate Mott- Glass phase [14]. 

Here V1 is the lattice potential deriving from all sites except that labelled 0, and the fts are 7r-electron wavefunctions for the unpaired electrons in the TCNQ ions. If we have one ion per unit cell (hl = h2 == h0), the energy is given by (33),... 

Figure 1.23 Movement of a domain wall in the lattice potential.

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